Shut down the good engine on a twin happens relatively often. So the pilots did a good thing only idling the assumed faulty one. Why did it never appear to them to give the idling engine a shot while they had already called for the coast guard. Task saturation? I would think you firewall all power you have if you go down over an ocean at night. Why stick to a wrong decision making if there is very little to loose if you try the other options? May be the training is not optimal for those situations?